A new finite difference stability criterion

A MAJOR DIFFICULTY in the numerical solution of partial differential equations of the parabolic or hyperbolic types by means of finite difference approximations is the tendency of the solution to be unstable under certain conditions. Thus, a small error (such as a round-off error), arising at some point in the computation procedure may tend to become larger and larger as the computation progresses, until the error terms completely overshadow the desired solution, making it worthless. Some finite difference approximations appear to be stable under any condition, others are always unstable, whereas some are stable only if the spacing intervals satisfy certain requirements.